How Did Socrates Teach the Boy to Double the Area of a Square?

Date: 06/15/2010 at 09:46:53
From: Anne
Subject: Doubling the area of a square

I was recently reading Meno by Plato. I was then asked to take the method
learned by the boy in the story, and use it to to double the area of a
square to quadruple the area of a square, or make one eight times as large.

I drew a 2x2 square, then followed the instructions given in Meno to
double the square: draw a 4x4 square ... but I'm not making the
connections. I think the method has something to do with the length of the
diagonal. But if you are just drawing squares, how can you find the length
of the diagonal? I keep resorting to the Pythagorean Theorem, but in Meno
there is no mention of it. Still, the boy gets the correct answer.

What am I missing? Can you explain the method to me or help me to see
the pattern?

Date: 06/15/2010 at 16:41:41
From: Doctor Rick
Subject: Re: Doubling the area of a square

Hi, Anne.

I located a translation here:

    http://classics.mit.edu/Plato/meno.html 

The terminology is a bit different from what we're used to, so it can be
hard to follow. But here's how I see it.

Socrates begins by presenting a square with sides of 2 feet, and eliciting
the fact that the area of the square is 4 square feet -- quadruple the
area of a square with sides of one foot. Then he asks how to double the
area, eliciting the guess that doubling the side will double the area. He
proceeds to help the boy see that this new square has four times the area
(16 square feet), not 8 square feet. The square sought must therefore have
sides greater than 2 feet but less than 4 feet. More Socratic methodology
elicits another guess that the sides must be 3 feet (halfway between). He
demonstrates that the area of this square (9 square feet) is still not the
desired 8 square feet.

All this is setup, bringing the boy around to acknowledge that he does not
know how to make a square with double the area. In the process, the boy
has made progress from thinking that he *did* know how to do it, when
he didn't.

At this point, Socrates constructs a figure like this, by making 4 copies
of the original 2-foot square:

  +-------+-------+
  |       |       |
  |       |       |2 feet
  |       |       |
  +-------+-------+
  |       |       |
  |       |       |2 feet
  |       |       |
  +-------+-------+
    2 feet  2 feet

This larger square (the first guess of the boy, actually) has an area 4
times that of the original square (one of the four small squares in the
figure), or 16 square feet. We're looking for a square with an area half
that size, 8 square feet.

Next Socrates constructs a diagonal in each square:

  +-------+-------+
  |     / | \     |
  |   /   |   \   |2 feet
  | /     |     \ |
  +-------+-------+
  | \     |     / |
  |   \   |   /   |2 feet
  |     \ | /     |
  +-------+-------+
    2 feet  2 feet

He points out that each diagonal divides its square into equal parts, each
part therefore half the area of the original. If we take one half of each
square, we get a region with half the area of the large square, or 8
square feet -- just what we're looking for:

          +
        /   \
      /       \
    /           \
  +               +
    \           /
      \       /
        \   /
          +

This is a square with twice the area of the original square.

Now, notice that there is no attempt to find the LENGTH of the diagonals
from which this square is constructed. This is typical of Greek
mathematics: their focus was not on numbers, but on geometric
construction. The Pythagorean Theorem can indeed be used to find the
length, and this theorem was indeed Greek, but the numeric aspect was
secondary in their thinking. In fact, they would express the Pythagorean
Theorem geometrically: "The [area of the] square constructed on the
hypotenuse of a right triangle is equal to the sum of the [areas of the]
squares constructed on the other two sides."

What Socrates and the boy did was to CONSTRUCT a square on the diagonal of
the given square, and to demonstrate that this square is twice the given
square (in area).

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/15/2010 at 16:59:26
From: Anne
Subject: Doubling the area of a square

How then can I apply this method to quadruple the area of a square or
reduce the area to one fourth the area of the original square?

Date: 06/15/2010 at 17:29:19
From: Doctor Rick
Subject: Re: Doubling the area of a square

Hi, Anne.

Quadrupling is easy: that's what came of one of the boy's guesses, when he
THOUGHT he was doubling! (Can you see which construction produced
this result?)

Reducing the area to 1/4 of the original is the reverse process of that.
Can you picture starting with the result and ending up with the square you
started with? What would that involve?

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/15/2010 at 21:13:55
From: Anne
Subject: Doubling the area of a square

I think I get it. If I multiply both dimensions by "a," then the area is
multiplied by "a^2." On the other hand, if I cut the side lengths in half,
then the area is multiplied by one half squared, or one fourth. Thus, by
doubling the side lengths the area gets multiplied by 2^2 or 4
(quadrupled), which is what the boy did.

I assume this always works?

Date: 06/15/2010 at 22:06:36
From: Doctor Rick
Subject: Re: Doubling the area of a square

Hi, Anne.

Yes, it does. If the sides of the original square are "s," then the area
is s^2. When you multiply the side length by "n," the side length is sn
and the area is (sn)^2 = s^2 n^2, or n^2 times the area of the
original square.

If you want to double the area, you want n^2 to equal 2. In other words,
you want n to be the square root of 2. That's what the diagonal of the
square is (according to the Pythagorean Theorem): the side of the square
multipied by the square root of 2.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/16/2010 at 21:11:26
From: Anne
Subject: Thank you (Doubling the area of a square)

Dr. Rick:

Thank you for your time and effort in helping me understand doubling the
area of a square as described in Meno. I've never used Dr. Math before,
but am truly pleased I did. I appreciate your timeliness and
thought-provoking questions.

Sincerely, 

Anne